Average Error: 13.1 → 0.7
Time: 5.7s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\sqrt[3]{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}} \cdot \sqrt[3]{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}}\right) \cdot \sqrt[3]{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}}\right) - 4.5\]
\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5
↓
\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\sqrt[3]{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}} \cdot \sqrt[3]{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}}\right) \cdot \sqrt[3]{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}}\right) - 4.5
double code(double v, double w, double r) {
	return (((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5);
}
↓
double code(double v, double w, double r) {
	return (((3.0 + ((2.0 / r) / r)) - ((cbrt(((0.125 * (3.0 - (2.0 * v))) * (pow(fabs((w * r)), 2.0) / (1.0 - v)))) * cbrt(((0.125 * (3.0 - (2.0 * v))) * (pow(fabs((w * r)), 2.0) / (1.0 - v))))) * cbrt(((0.125 * (3.0 - (2.0 * v))) * (pow(fabs((w * r)), 2.0) / (1.0 - v)))))) - 4.5);
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.1

    \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt13.1

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r} \cdot \sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}\right)}}{1 - v}\right) - 4.5\]

  4. Simplified13.1

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left|w \cdot r\right|} \cdot \sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}\right)}{1 - v}\right) - 4.5\]

  5. Simplified7.1

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left|w \cdot r\right| \cdot \color{blue}{\left|w \cdot r\right|}\right)}{1 - v}\right) - 4.5\]
  6. Using strategy rm

  7. Applied *-un-lft-identity7.1

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right)}{\color{blue}{1 \cdot \left(1 - v\right)}}\right) - 4.5\]

  8. Applied times-frac0.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1} \cdot \frac{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}{1 - v}}\right) - 4.5\]

  9. Simplified0.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right)} \cdot \frac{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}{1 - v}\right) - 4.5\]

  10. Simplified0.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}}\right) - 4.5\]
  11. Using strategy rm

  12. Applied associate-/r*0.4

    \[\leadsto \left(\left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}\right) - 4.5\]
  13. Using strategy rm

  14. Applied add-cube-cbrt0.7

    \[\leadsto \left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \color{blue}{\left(\sqrt[3]{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}} \cdot \sqrt[3]{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}}\right) \cdot \sqrt[3]{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}}}\right) - 4.5\]

  15. Final simplification0.7

    \[\leadsto \left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\sqrt[3]{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}} \cdot \sqrt[3]{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}}\right) \cdot \sqrt[3]{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{{\left(\left|w \cdot r\right|\right)}^{2}}{1 - v}}\right) - 4.5\]

Reproduce

herbie shell --seed 2020049 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))